This disclosure relates generally to the field of determining time and position of origin of seismic events occurring in the subsurface. More particularly, the disclosure relates to techniques for determining uncertainty in the determined positions and times of origin of such seismic events.
In passive seismic surveying, sensors (e.g., geophones) are deployed to record seismic response at various locations. A set of possible subsurface seismic event (source) locations are defined, in one example case a 3D grid of points presumably encompassing all event location. For each point in this set, for a travel time from presumed source location to each sensor, the recorded data from each sensor is time shifted to remove the travel time delay, then the time shifted responses from all sensors are summed. For a given time span, local peaks in the summed response are determined among the set of possible source locations. The locations and timing of these peaks are taken as estimates of the location and origin time of the seismic events. An example technique for determining estimated time of origin and position of the seismic events is described in U.S. Pat. No. 7,663,970 issued to Duncan et al. and incorporated herein by reference in its entirety.
One problem in microseismic data analysis is to estimate some set of parameters of interest from data collected during an experiment. A maximum-likelihood estimator is a mathematical process that produces an estimate of a set of model parameters by finding the maximum probability (likelihood) of given data. The likelihood function is constructed from a statistical description of the noises present in the data, a mathematical model of the data generation process and the data. Once this likelihood function is specified, an estimate of the parameters may be obtained by application of an appropriate optimization strategy, to determine the values of the parameters that maximize the likelihood.
Using concepts from estimation theory it is possible to compute estimates of the uncertainty in the estimates obtained from a maximum likelihood estimator. An estimate of the uncertainty can be obtained following a process known as the “Cramer-Rao lower-bound”. Here the variance of the estimator (Var({circumflex over (ξ)}) can be shown to be bounded below by the values given by elements of the inverse of the Fisher Information Matrix (F).
{circumflex over (ξ)}=MLE of parametersVar({circumflex over (ξ)}1)≧[Fii]−1 
The Fisher information matrix is computed from the second derivatives of the natural log of the likelihood function:
            F      ij        =          -                                    ∂            2                    ⁢                      ℒ            ⁡                          (              ξ              )                                                            ∂                          ξ              i                                ⁢                      ∂                          ξ              j                                                      ℒ      ⁡              (        ξ        )              =          ln      ⁡              (                  likelihood          ⁡                      (                          ξ              ❘              data                        )                          )            